Hints and Solutions: Patty Paper Trisection

No peeking! This post is for those of you who have given the trisection proof a good workout on your own. If you have a question about the proof or a solution you would like to share, please post a comment here.

But if you haven’t yet worked at the puzzle, go back and give it a try. When someone just tells you the answer, you miss out on the fun. Figure it out for yourself — and then check the answer just to prove that you got it right.

The Bestselling Math Book of All Time

When all else fails, ask a teacher — and for proofs, who better than Euclid? His geometry textbook, the Elements, is the bestselling math textbook of all time. Here are some tips from the master:

Book 1, Proposition 5
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.

Book 1, Proposition 29
A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the sum of the interior angles on the same side equal to two right angles.

Book 1, Proposition 32
In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.

Book 1, Proposition 33
Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel.

That “Aha!” Feeling

When I tried this proof for myself, I thought it was relatively easy to prove 2 of the small angles congruent. But to show that they were both equal to the third angle was tough — and unless I could show that all 3 of the small angles were the same, my proof wasn’t worth anything. As I fidgeted with my patty paper, I noticed that two lines seemed to overlap.

(Click the image for a larger view.)

My geometry teacher in high school drilled one rule repeatedly, until it was permanently embedded in my brain: Never trust the drawing! But what if it wasn’t just a coincidence? What if the lines really did match?

Aha! That was my key. As soon as I could show that those two lines would always coincide, no matter what the original angle was, the rest of my proof was a cinch.

Copyright Notice

Everything on this website, like nearly everything else on the Internet, is copyrighted material. It is illegal (and terribly rude) to take what another blogger has written and paste it into your own blog or website.

I hope you enjoy my tips and math games, use them in your classroom or homeschool, mention them to friends, link to them as much as you want‌—‌but please respect my copyrights, as you would want me to respect yours.

Affiliate Disclosure

Almost all of the book covers featured on this site link to Amazon.com, where you can read descriptions and reviews‌—‌and where I earn a small commission if you actually buy a book. But if you have access to a good library loan system, you should be able to read most of the books for free.